3 Friedmann Robertson Walker Universe

Transcription

1 28 3 Friedmann Robertson Walker Universe 3. Kinematics 3.. Robertson Walker metric We adopt now the cosmological principle, and discuss the homogeneous and isotropic model for the universe. This is called the Friedmann Robertson Walker (FRW) or the Friedmann Lemaître Robertson Walker universe. Here the homogeneity refers to spatial homogeneity only, so that the universe will still be different at different times. Spatial homogeneity means that there exists a coordinate system whose t = const hypersurfaces are homogeneous. This time coordinate is called the cosmic time. Thus the spatial homogeneity property selects a preferred slicing of the spacetime, reintroducing a separation between space and time. There is good evidence, that the Universe is indeed rather homogeneous (all places look the same) and isotropic (all directions look the same) at sufficiently large scales (i.e., ignoring smaller scale features), larger than Mpc. (Recall the discussion of the cosmological principle in Chapter.) Homogeneity and isotropy mean that the curvature of spacetime must be the same everywhere and into every space direction, but it may change in time. It can be shown that the metric can then be given (by a suitable choice of the coordinates) in the form [ ds 2 = 2 +a 2 dr 2 ] (t) Kr 2 +r2 dϑ 2 +r 2 sin 2 ϑdϕ 2, () the Robertson Walker (RW) metric in spherical coordinates. Doing a coordinate transformation we can also write it in Cartesian coordinates (exercise): ds 2 = 2 +a 2 dx 2 +dy 2 +dz 2 (t)[ + 4 K(x2 +y 2 +z 2 ) ] 2, (2) where x = rsinϑcosϕ, y = rsinϑsinϕ, z = rcosϑ, and r = ( Kr 2 )/( 2Kr). Usually the spherical coordinates are more useful. This is thus the metric of our universe, to first approximation, and we shall work with this metric for a large part of this course. The time coordinate t is the cosmic time. Here K is a constant, related to curvature of space, and a(t) is a function of time, related to expansion (or possible contraction) of the universe. We call R curv a(t)/ K (3) the curvature radius of space (at time t). The time-dependent factor a(t) is called the scale factor. When the Einstein equation is applied to the RW metric, we will get the Friedmann equations, from which we can solve a(t). This will be done in Sec For now, a(t) is an arbitrary function of the time coordinate t. However, for much of the following discussion, we will assume that a(t) grows with time; and when we refer to the age of the universe, we assume that a(t) becomes zero at some finite past, which we take as the origin of the time coordinate, t =. That is, for the whole of Cosmology I. In Cosmology II we shall consider deviations from this homogeneity. 2 I have adopted from Syksy the separation of this chapter into Kinematics (Sec. 3.: RW metric only) and Dynamics (Sec. 3.2: RW metric + Friedmann equations). This has the advantage that this Kinematics section applies also to other metric theories of gravity than general relativity, which one may want to consider at some point.

2 3 FRIEDMANN ROBERTSON WALKER UNIVERSE 29 We use the dot, d/, to denote derivatives with respect to cosmic time t and define H ȧ/a. (4) This quantity H = H(t) gives the expansion rate of the universe, and it is called the Hubble parameter. Its present value H is the Hubble constant. (In cosmology it is customary to denote the present values of quantities with the subscript.) The dimension of H is /time (or velocity/distance). In time a distance gets stretched by a factor +H (a distance L grows with velocity HL). Note that although the metric describes a homogeneous universe, the metric itself is not explicitly homogeneous, because it depends also on the coordinate system in addition to the geometry. (This is a common situation, just like the spherical coordinates of a sphere do not form a homogeneous coordinate system, although the sphere itself is homogeneous.) However, any physical quantities that we calculate from the metric are homogeneous and isotropic. We notice immediately that the 2-dimensional surfaces t = r = const have the metric of a sphere with radius ar. Since the universe is homogeneous, the location of the origin (r = ) in space can be chosen freely. We naturally tend to put ourselves at the origin, but for calculations this freedom may be useful. We have the freedom to rescale the radial coordinate r. For example, we can multiply all values of r by a factor of 2, if we also divide a by a factor of 2 and K by a factor of 4. The geometry of the spacetime stays the same, just the meaning of the coordinate r has changed: the point that had a given value of r has now twice that value in the rescaled coordinate system. There are two common ways to rescale:. If K, we can rescale r to make K equal to ±. In this case K is usually denoted k, and it has three possible values, k =,,+. In this case r is dimensionless, and a(t) has the dimension of distance. For k = ±, a(t) becomes equal to R curv and is often denoted R(t). Equations in this convention will be written in blue. 2. The other way is to rescale a to be one at present 3, a(t ) a =. In this case a(t) is dimensionless, whereas r and K /2 have the dimension of distance. We will adopt this convention from Sec on. Choosing one of these two scalings will simplify some of our equations. One must be careful about the possible confusion resulting from comparing equations using different scaling conventions. IfK =, thespacepart(t =const) of therobertson Walker metricisflat. The3-metric (the space part of the full metric) is that of ordinary Euclidean space written in spherical coordinates, with the radial distance given by ar. The spacetime, however, is curved, since a(t) depends on time, describing the expansion or contraction of space. In common terminology, we say the universe is flat in this case. If K >, the coordinate system is singular at r = / K. (Remember our discussion of the 2-sphere!) With the substitution (coordinate transformation) r = K /2 sin(k /2 χ) the metric becomes [ ds 2 = 2 +a 2 (t) dχ 2 +K sin 2 (K /2 χ)dϑ 2 +K sin 2 (K /2 χ)sin 2 ϑdϕ 2]. (5) With the scaling choice K = k = this simplifies to ds 2 = 2 +a 2 (t) [ dχ 2 +sin 2 χdϑ 2 +sin 2 χsin 2 ϑdϕ 2]. (6) 3 In some discussions of the early universe, it may also be convenient to rescale a to be one at some particular early time.

3 3 FRIEDMANN ROBERTSON WALKER UNIVERSE 3 Figure : The hypersphere. This figure is for K = k =. Consider the semicircle in the figure. It corresponds to χ ranging from to π. You get the (2-dimensional) sphere by rotating this semicircle off the paper around the vertical axis by an angle ϕ = 2π. You get the (3-dimensional) hypersphere by rotating it twice, in two extra dimensions, by ϑ = π and by ϕ = 2π, so that each point makes a sphere. Thus each point in the semicircle corresponds to a full sphere with coordinates ϑ and ϕ, and radius (a/ K)sinχ. The space part has the metric of a hypersphere (a 3-sphere), a sphere with one extra dimension. Kχ is a new angular coordinate, whose values range over π, just like ϑ. The singularity at r = / K disappears in this coordinate transformation, showing that it was just a coordinate singularity, not a singularity of the spacetime. The original coordinates covered only half of the hypersphere, as the coordinate singularity r = / K divides the hypersphere into two halves. The case K > corresponds to a closed universe, whose (spatial) curvature is positive. 4 This is a finite universe, with circumference 2πa/ K = 2πR curv and volume 2π 2 K 3/2 a 3 = 2π 2 Rcurv, 3 and we can think of R curv as the radius of the hypersphere. If K <, we do not have a coordinate singularity, and r can range from to. The substitution r = K /2 sinh( K /2 χ) is, however, often useful in calculations. The case K < corresponds to an open universe, whose (spatial) curvature is negative. The metric is then [ ds 2 = 2 +a 2 (t) dχ 2 + K sinh 2 ( K /2 χ) ( dϑ 2 +sin 2 ϑdϕ 2)]. (7) This universe is infinite, just like the case K =. To handle all three curvature cases simultaneously, we define K /2 sin(k /2 χ), (K > ) f K (χ) χ, (K = ) K /2 sinh( K /2 χ), (K < ) (8) 4 Positive (negative) curvature means that the sum of angles of any triangle is greater than (less than) 8 and that the area of a sphere with radius χ is less than (greater than) 4πχ 2.

4 3 FRIEDMANN ROBERTSON WALKER UNIVERSE 3 which allows us to write the RW metric as [ ds 2 = 2 +a 2 (t) dχ 2 +fk(χ) 2 ( dϑ 2 +sin 2 ϑdϕ 2)], (9) The RW metric (at a given time) has two associated length scales. The first is the curvature radius, R curv a K /2. The second is given by the time scale of the expansion, the Hubble time or Hubble length t H l H H, whose present value is H = 9.778h Gyr = h Mpc. () (Note that due to the definition of h, the digits is just the speed of light in units of km/s, which makes this value of l H easy to remember.) In the case K = the universe is flat, so the only length scale is the Hubble length. The coordinates (t,r,ϑ,ϕ) or (t,x,y,z) of the RW metric are called comoving coordinates. This means that the coordinate system follows the expansion of space, so that the space coordinates of objects which do not move remain the same. The homogeneity of the universe fixes a special frame of reference, the cosmic rest frame given by the above coordinate system, so that, unlike in special relativity, the concept does not move has a specific meaning. The coordinate distance between two such objects stays the same, but the physical, or proper distance between them grows with time as space expands. The time coordinate t, the cosmic time, gives the time measured by such an observer at rest, at (r,ϑ,ϕ) = const. It can be shown that the expansion causes the motion of an object in free fall to slow down with respect to the comoving coordinate system. For nonrelativistic physical velocities v, v(t 2 ) = a(t ) a(t 2 ) v(t ). () The peculiar velocity of a galaxy is its velocity with respect to the comoving coordinate system Redshift Let us now ignore the peculiar velocities of galaxies (i.e., we assume they are = ), so that they will stay at fixed coordinate values (r,ϑ,ϕ), and find how their observed redshift z arises. We set the origin of our coordinate system at galaxy O (observer). Let the r-coordinate of galaxy A be r A. Since we assumed the peculiar velocity of galaxy A to be, the coordinate r A stays constant with time. Light leaves the galaxy at time t with wavelength λ and arrives at galaxy O at time t 2 with wavelength λ 2. It takes a time δt = λ /c = /ν to send one wavelength and a time δt 2 = λ 2 /c = /ν 2 to receive one wavelength. Follow now the two light rays sent at times t and t +δt (see figure). Along the light rays t and r change, ϑ and ϕ stay constant (this is clear from the symmetry of the problem). Light obeys the lightlike condition We have thus ds 2 =. (2) ds 2 = 2 +a 2 dr 2 (t) Kr 2 = 2 +a 2 (t)dχ 2 = (3) dr = = dχ. a(t) Kr 2 (4)

5 3 FRIEDMANN ROBERTSON WALKER UNIVERSE 32 Figure 2: The two light rays to establish the redshift. Integrating this, we get for the first light ray, and for the second, t2 t t2 +δt 2 t +δt a(t) = a(t) = ra ra dr Kr 2 = χa dr Kr 2 = χa dχ = χ A, (5) dχ = χ A. (6) The right hand sides of the two equations are the same, since the sender and the receiver have not moved (they have stayed at r = r A and r = ). Thus = t2 +δt 2 t +δt a(t) t2 t a(t) = and the time to receive one wavelength is t2 +δt 2 t 2 a(t) t +δt t a(t) = δt 2 a(t 2 ) δt a(t ), (7) δt 2 = a(t 2) a(t ) δt. (8) As is clear from the derivation, this cosmological time dilation effect applies to observing any event taking place in galaxy A. As we observe galaxy A, we see everything happening in slow motion, slowed down by the factor a(t 2 )/a(t ), which is the factor by which the universe has expanded since the light (or any electromagnetic signal) left the galaxy. This effect can be observed, e.g., in the light curves of supernovae (their luminosity as a function of time). For the redshift we get +z λ 2 = δt 2 = a(t 2) λ δt a(t ). (9) The redshift of a galaxy directly tells us how much smaller the universe was when the light left the galaxy. The result is easy to remember; the wavelength expands with the universe. Thus the redshift z is related to the value of a(t) and thus to the time t, the age of the universe, when the light left the galaxy. We can thus use a or z as alternative time coordinates. Their relation is +z = a a or a = a +z da a = dz +z da = a dz (+z) 2. (2)

6 3 FRIEDMANN ROBERTSON WALKER UNIVERSE 33 Note that while a grows with time, z decreases with time: z = at a = t = and z = at t = t Age-redshift relation If the observed redshift of a galaxy is z, what was the age of the universe when the light left the galaxy? Without knowing the function a(t) we cannot answer this and other similar questions, but it is useful to derive general expressions in terms of a(t), z, and H(t). We get H = da a so that the age of the universe at redshift z is t(z) = and the present age of the universe is = da ah = dz H(+z), (2) t = t t(z = ) = z H(+z ). (22) H(+z ). (23) The difference gives the light travel time, i.e., how far in the past we see the galaxy, 3..4 Distance t t(z) = z H(+z ). (24) In cosmology, the typical velocities of observers (with respect to the comoving coordinates) are small, v < km/s, so that we do not have to worry about Lorentz contraction (or about the velocity-related time dilation) and in the FRW model we can use the cosmic rest frame. The expansion of the universe brings, however, other complications to the concept of distance. Do we mean by the distance to a galaxy how far it is now (longer), how far it was when the observed light left the galaxy (shorter), or the distance the light has traveled (intermediate)? The proper distance (or physical distance ) d p (t) between two objects 5 is defined as their distance measured along the hypersurface of constant cosmic time t. By comoving distance we mean the proper distance scaled to the present value of the scale factor (or sometimes to some other special time we choose as the reference time). If the objects have no peculiar velocity their comoving distance at any time is the same as their proper distance today. To calculate the proper distance d p (t) between galaxies (one at r =, another at r = r A ) at time t, we need the metric, since d p (t) = r A ds. We integrate along the path t,ϑ,ϕ = const, or = dϑ = dϕ =, so ds 2 = a 2 (t)dχ 2 = a 2 (t) dr 2 Kr2, and get ra d p dr χa (t) = a(t) = a(t) dχ Kr 2 K /2 a(t)arcsin(k /2 r A ) (K > ) = a(t)r A (K = ) K /2 a(t)arsinh( K /2 r A ) (K < ) a(t)f K (r A) = a(t)χ A (25) 5 or more generally between two points (r,ϑ,ϕ) on the t = const hypersurface. In relativity, proper length of an object refers to the length of an object in its rest frame, so the use of the word proper in proper distance is perhaps proper only when the objects are at rest in the FRW coordinate system. Nevertheless, we define it now this way.

7 3 FRIEDMANN ROBERTSON WALKER UNIVERSE 34 Figure 3: Calculation of the distance-redshift relation. The functions f K (χ) and r fk (r) dr = Kr 2 K /2 arcsin(k /2 r), (K > ) r, (K = ) K /2 arsinh( K /2 r). (K < ) (26) convert between the two natural unscaled (i.e., you still need to multiply this distance by the scale factor a) radial distance definitions for the RW metric: the proper distance measured along the radial line, and χ = fk dp (r) = a, (27) r = f K (χ) = f K (d p /a) (28) which is related to the length of the circle and the area of the sphere at this distance with the familiar 2πar and 4π(ar) 2. As the universe expands, the proper distance grows, d p (t) = a(t)χ = a +z χ dc +z, (29) where d c a χ = d p (t ) is the present proper distance to r, or the comoving distance to r. We adopt now the scaling convention a =, so that the coordinate χ becomes equivalent to the comoving distance from the origin. The comoving distance between two different objects, A and B, lying along the same line of sight, i.e., having the same ϑ and ϕ coordinates, is simply χ B χ A. Both f K and f K have the dimension of distance. Neither the proper distance d p, nor the coordinate r of a galaxy are directly observable. Observable quantities are, e.g., the redshift z, location on the sky (ϑ,ϕ) when the observer is at r =, the angular diameter, and the apparent luminosity. We want to use the RW metric to relate these observable quantities to the coordinates and actual distances. Let us first derive the distance-redshift relation. See Fig. 3. We see a galaxy with redshift z; how far is it? (We assume z is entirely due to the Hubble expansion, +z = /a, i.e., we ignore the contribution from the peculiar velocity of the galaxy or the observer). Since for light, ds 2 = 2 +a 2 dr 2 (t) Kr 2 = 2 +a 2 (t)dχ 2 =, (3)

8 3 FRIEDMANN ROBERTSON WALKER UNIVERSE 35 we have dr t = a(t) = a(t)dχ Kr 2 The comoving distance to redshift z is thus d c (z) = χ(z) = t t a(t) = t a(t) = +z The proper distance at the time the light left the galaxy is d p (z) = +z z da a r z da/ = dr Kr 2 = χ = dc. (3) H(z ). (32) H(z ). (33) The distance light has traveled (i.e., adding up the infinitesimal distances measured by a sequence of observers at rest along the light path) is equal to the light travel time, Eq. (24). In a monotonously expanding (or contracting) universe it is intermediate between d p (z) and d c (z). We encounter the beginning of time, t =, at a = or z =. Thus the comoving distance light has traveled during the entire age of the universe is d c hor = χ hor = H(z ). (34) This distance (or the sphere with radius d c hor, centered on the observer) is called the horizon, since it represent the maximum distance we can see, or receive any information from. There are actually several different concepts in cosmology called the horizon. To be exact, the one defined above is the particle horizon. Another horizon concept is the event horizon, which is related to how far light can travel in the future. The Hubble distance H is also often referred to as the horizon (especially when one talks about subhorizon and superhorizon distance scales) Volume Theobjects we observe lie on our past light cone, and the observed quantities are z,ϑ,ϕ, so these are the observer s coordinates for the light cone. What is the volume of space corresponding to a range z ϑ ϕ? Note that the light cone is a lightlike surface, so its volume is zero. Here we mean instead the volume that we get when we project a section of it onto the t = const hypersurface crossing this section at a particular z (which is unique when z = dz is infinitesimal). From Eq. (33), the proper distance corresponding to dz is dz/[( + z)h(z)]. Directly from the RW metric, the area corresponding to dϑdϕ is ardϑ arsinϑdϕ, so that the proper volume element becomes dv p = a2 r 2 sinϑ (+z)h(z) dzdϑdϕ = a 2 r 2 dzdω (35) (+z)h(z) and the comoving volume is dv c = (+z) 3 dv = r2 dzdω. (36) H(z) These are the volume elements for counting the number density or comoving number density of galaxies from observations. If the number of galaxies is conserved, in a homogeneous universe their comoving number density should be independent of z. Thus, in principle, from such observations one should be able to determine H(z). In practice this is made difficult by evolution of galaxies with time, mergers of galaxies, and the fact that it is moredifficult to observe galaxies at larger z.

9 3 FRIEDMANN ROBERTSON WALKER UNIVERSE 36 Figure 4: Defining the angular diameter distance Angular diameter distance The distance-redshift relation (32) obtained above would be nice if we already knew the function a(t). We can turn the situation around and use an observed distance-redshift relation, to obtain information about a(t), or equivalently, about H(z). But for that we need a different distanceredshift relation, one where the distance is replaced by some directly observable quantity. Astronomers employ various such auxiliary distance concepts, like the angular diameter distance or the luminosity distance. These would be equal to the true distance in Euclidean non-expanding space. To answer the question: what is the physical size s of an object, whom we see at redshift z subtending an angle ϑ on the sky? we need the concept of angular diameter distance d A. In Euclidean geometry (see Fig. 4), Accordingly, we define s = ϑd or d = s ϑ. (37) d A sp ϑ, (38) wheres p was theproperdiameter of theobject whenthelight wesee left it, andϑis theobserved angle. For large-scale structures, which expand with the universe, we use the comoving angular diameter distance d c A sc /ϑ, where s c = ( + z)s p is the comoving diameter of the structure and z is its redshift. Thus d c A = (+z)d A. From the RW metric, the physical length s p corresponding to an angle ϑ is, from ds 2 = a 2 (t)r 2 dϑ 2 s p = a(t)rϑ. Thus d A (z) = a(t)r = r +z = f K(χ) +z The comoving angular diameter distance is then ( d c A = r = f da K a +z da/ = = ) ( +z f K ( z +z f K = f K ( z +z da a H(z ) ) da/ ) (39) ) H(z. (4) ) For the flat (K = ) FRW model r = f K (χ) = χ, so that the angular diameter distance is equal to the proper distance when the light left the object and the comoving angular diameter distance is equal to the comoving distance. For large distances (redshifts) the angular diameter distance may have the counterintuitive property that after some z it begins to decrease as a function of z. Thus objects with are behind other objects as seen from here will nevertheless have a smaller angular diameter distance. There are two reasons for such behavior: In a closed (K > ) universe objects which are on the other side of the universe (the 3-sphere), i.e., with K /2 χ > π/2, will cover a larger angle as seen from here because of the

10 3 FRIEDMANN ROBERTSON WALKER UNIVERSE 37 spherical geometry (if we can see this far). This effect comes from the f K in Eq. (39). An object at exactly opposite end (K /2 χ = π) would cover the entire sky as light from it would reach us from every direction after traveling half-way around the 3-sphere. In our universe these situations do not occur in practice, because lower limits to the size of the 3-sphere 6 are much larger than the distance light has traveled in the age of the Universe. The second reason, which does apply to the observed universe, and applies only to d A, not to d c A, is the expansion of the universe. An object, which does not expand with the universe, occupied a much larger comoving volume in the smaller universe of the past. This effect is the /(+z) factor in Eq. (39), which for large z decreases faster than the other part grows. In other words, the physical size of the 2-sphere corresponding to a given redshift z has a maximum at some finite redshift (of the order z ), and for larger redshifts it is again smaller. (The same behavior applies to the proper distance d p (z).) Suppose we have a set of standard rulers, objects that we know are all the same size s p, observed at different redshifts. Their observed angular sizes ϑ(z) then give us the observed angular diameter distance as d A (z) = s p /ϑ(z). This observed function can be used to determine the expansion history a(t), or H(z) Luminosity distance In transparent Euclidean space, an object whose distance is d and whose absolute luminosity (radiated power) is L would have an apparent luminosity l = L/4πd 2. Thus we define the luminosity distance of an object as L d L 4πl. (4) Consider the situation in the RW metric. The absolute luminosity can be expressed as: L = number of photons emitted time their average energy = N γe em t em. (42) If the observer (at present time, a = ) is at a coordinate distance r from the source (note how we now put the origin of the coordinate system at the source), the photons have at that distance spread over an area A = 4πr 2. (43) The apparent luminosity can be expressed as: l = number of photons observed time area their average energy = N γe obs t obs A. (44) The number of photons N γ is conserved, but their energy is redshifted, E obs = E em /( + z). Also, if the source is at redshift z, it takes a factor +z longer to receive the photons t obs = (+z)t em. Thus, l = N γe obs t obs A = N γe em t em (+z) 2 4πr 2. (45) From Eq. (4), d L L 4πl = (+z)r = (+z)d c A(z) = (+z) 2 d A (z). (46) Compared to the comoving angular diameter distance, d c A (z), we have a factor (+z), which causes d L to increase faster with z than d c A (z). There is one factor of ( + z)/2 from photon 6 Observations tell us that the curvature of the Universe is very small, so that the we have so for not been able to determine which of the three geometries applies to it.

11 3 FRIEDMANN ROBERTSON WALKER UNIVERSE 38 redshift and another factor of (+z) /2 from cosmological time dilation, both contributing to making large-redshift objects dimmer. Compared to d A (z), thereis another factor of (+z) from the expansion of the universe, which we discussed in Sec. 3..6, which causes distant objects to appear larger on the sky, but does not contribute to their apparent luminosity. Thus the surface brightness (flux density per solid angle) of objects decreases with redshift as d 2 A /d2 L = (+z) 4 (47) (flux density l d 2 L, solid angle Ω d 2 A.) Suppose that we have a set of standard candles, objects that we know all have the same L. From their observed redshifts and apparent luminosities we get an observed luminosity-distanceredshift relation d L (z) = L/4πl, which can be used to determine a(t), or H(z) Hubble law In Sec. we introduced the Hubble law z = H d d = H z, (48) which was based on observations (at small redshifts). Now that we have introduced the different distance concepts, d p, d c, d A, d c A, d L, in an expanding universe, and derived exact formulae (33, 32, 39, 4, 46) for them in the RW metric, we can see that for z (when we can approximate H(z) = H ) all of them give the Hubble law as an approximation, but all of them deviate from it, in a different manner, for z and larger Conformal time In the comoving coordinates of Eqs. (), (6), and (7), the space part of the coordinate system is expanding with the expansion of the universe. It is often practical to make a corresponding change in the time coordinate, so that the unit of time (i.e., separation of time coordinate surfaces) also expands with the universe. The conformal time η is defined by dη t a(t), or η = a(t ). (49) The RW metric acquires the form ] ds 2 = a 2 (η) [ dη 2 + dr2 Kr 2 +r2( dϑ 2 +sin 2 ϑdϕ 2), (5) or with the other choice of the radial coordinate, χ, [ ds 2 = a 2 (η) dη 2 +dχ 2 +fk(χ) 2 ( dϑ 2 +sin 2 ϑdϕ 2)], (5) The form (5) is especially nice for studying radial (dϑ = dϕ = ) light propagation, because the lightlike condition ds 2 = becomes dη = ±dχ. In the end of the calculation one may need to convert conformal time back to cosmic time to express the answer in terms of the latter.